Cyclotomic Completions of Polynomial Rings

Abstract

For a subset S ⊂ ℕ = {1, 2, . . . } and a commutative ring R with unit, let R[q]S denote the completion lim←_f_(q) R[q]/(f(q)), where f(q) runs over all the products of the powers of cyclotomic polynomials Φ_n_(q) with n ∈ S. We will show that under certain conditions the completion R[q]S can be regarded as a “ring of analytic functions” defined on the set of roots of unity of order in S. This means that an element of R[q]S vanishes if it vanishes on a certain type of infinite set of roots of unity, or if its power series expansion at one root of unity vanishes. In particular, the completion ℤ[q]N ≃ lim←_n_ ℤ[q]/((1 − q)(1 − _q_2 ) · · · (1 − qn)) enjoys this property.